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  • Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Volume 126, Issue 2
  • January 1996, pp. 297-342

Multiscale convergence and reiterated homogenisation

  • G. Allaire (a1) and M. Briane (a2) (a3)
  • DOI: http://dx.doi.org/10.1017/S0308210500022757
  • Published online: 14 November 2011
Abstract

This paper generalises the notion of two-scale convergence to the case of multiple separated scales of periodic oscillations. It allows us to introduce a multi-scale convergence method for the reiterated homogenisation of partial differential equations with oscillating coefficients. This new method is applied to a model problem with a finite or infinite number of microscopic scales, namely the homogenisation of the heat equation in a composite material. Finally, it is generalised to handle the homogenisation of the Neumann problem in a perforated domain.

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1E. Acerbi , V. Chiado Piat , G. Dal Maso and D. Percivale . An extension theorem from connected sets, and homogenization in general periodic domains. Nonlinear Anal. 18(1992). 481–5.

2G. Allaire . Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 (1992), 1482–518.

14E. Cabib and B. Dal Maso . On a class of optimum problems in structural design. J. Opiim. Theory Appl. 56 (1988). 3965.

15D. Cioranescu and J. Saint Jean Paulin . Homogenization in open sets with holes. J. Math. Anal. Appi. 71 (1979), 590607.

16G. Dal Maso . An Introduction to Gamma-convergence (Boston: Birkhauser, 1993).

29V. Zhikov , S. Kozlov , O. Oleinik and N. Ngoan . Averaging and G-convergence. Russian Math. Surveys 34 (1979), 69147.

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Proceedings of the Royal Society of Edinburgh Section A: Mathematics
  • ISSN: 0308-2105
  • EISSN: 1473-7124
  • URL: /core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics
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